Introduction

We have learnt that the MGF for a binimial random variable is:

\[M_X(t) = (q+pe^t)^n \]

Where \(X\sim Bin(n,p)\)

Tasks

Using the moment generating function above prove that

\[\sigma^2 = npq\]

by answering the tasks below. Use \(\LaTeX\) to construct the proof for Task 1.

Task 1

Start with

\[ \begin{eqnarray} E(X) &=& \left . \frac{d M_X(t)}{dt}\right |_{t=0}\\ &=& npe^t(q + pe^t)^{n - 1}\\ \end{eqnarray} \]

Task 2

You may use paper and write neatly the proof - take a picture and place in the document using

![](){}

Now find \(E(X^2)\)

\[ \begin{eqnarray} E(X^2) &=& \left . \frac{d^2 M_X(t)}{d^2t}\right |_{t=0}\\ &=& \end{eqnarray} \] “Task 2”

Task 3

Find

\(\sigma^2\) Using the formula \(\sigma^2 =E(X^2)-\mu^2\) “Task 3”